I'd love your help with this question.
Let $n\geq3$ be a fixed integer. How many non-isomorphic graphs with $V$ vertices and $E$ edges are there where $V+E=n$?
Thank you very much.
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1Do you have any reason to think there will be any kind of useful formula for this? – Gerry Myerson Jun 17 '11 at 00:55
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This is very hard for fixed $p$ and $q$, and $n$ isn't a particularly natural parameter here. – Qiaochu Yuan Jun 17 '11 at 00:56
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8For what it's worth, this is tabulated at http://oeis.org/A120412 – Gerry Myerson Jun 17 '11 at 01:00
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2FWIW, this was also posted at http://mathoverflow.net/questions/68017/counting-graphs – JavaMan Jun 17 '11 at 02:29
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Might a starting point be to think about how many planar connected graphs there which obey this condition? – Joseph Malkevitch Sep 01 '11 at 21:05
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This MSE link shows how to compute the generating function of non-isomorphic graphs on some number of nodes by the number of edges. The answer to your query is obtained by extracting the relevant coeffcients from these generating functions. – Marko Riedel Jun 12 '14 at 00:06
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This type of enumeration can be achieved using Polya Theory to attain a generating function. The best reference is Riordan, http://books.google.com/books?id=zWgIPlds29UC&lpg=PP1&pg=PP1#v=onepage&q&f=false, page 129. See page 143 for the application to counting graphs by |E| and |V|.
If would like to construct such graphs , there is a C program "nauty" developed by Brendan D. McKay, based on his math article http://cs.anu.edu.au/~bdm/nauty/pgi.pdf.
Gus Wiseman
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